A brief introduction to Combinatorial Games Asaf Ferber ∗ September 26, 2019 In these notes we give a brief introduction to combinatorial games. I used these notes few years ago for a mini-course that I gave for undergrads in MIT during the winter break (three lectures). Feel free to use/distribute it, and if you find errors/improve the presentation I would love to get an updated TEX file! (asaff@uci.edu). 1 Introduction How can one analyze a game? any attempt to analyze games like Chess, Go, Checkers, Tic-Tac- Toe, Hex etc. lead to the conclusion: there are enormously many cases to analyze and it seems quite hopeless. The only positive side is that it keeps the game interesting for a competition. As opposed to traditional Game Theory which focuses on incomplete information games (like Poker), here we consider perfect information games which are more similar in nature to Chess and Go. The term \Combinatorial Game" means a skill game, with no chance moves, where each player has a complete information about the current/past positions and can, theoretically, analyze all possible future position. The payoff function has 3 values: win, draw and loss. The first question one should ask: If these games are all deterministic, why can't we just use a computer to analyze? the simple answer is that even though in theory you can, practically there are just too many possibilities for any computer to analyze in a reasonable time. For example, consider a 3 dimensional version of the kids game Tic-Tac-Toe which is played on a board of size 5 × 5 × 5. This game has roughly 3125 positions (each position can be either marked \Player I", \Player II" or \Unoccupied"), which even though is only a finite number, it is so large that we would be afraid to meet it in a dark street at night. To summarize { traditional Game Theory is not useful for us, and a brute force computer search doesn't help even for very simple games as the 3 dimensions Tic-Tac-Toe mentioned above. What else can we do? 2 Examples and basic techniques In this section we discuss some examples and illustrate some useful techniques in analyzing certain games. ∗Department of Mathematics, UCI. Email: asaff@uci.edu. 1 2.1 Solitaire army The common feature of the Solitaire puzzles is that each one is played with a board and soldiers, the board contains a number of holes, each of which can hold 1 soldier. Each move consists of a jump by 1 soldier over 1 or more other soldiers. The soldiers jumped over being removed from the board. Each move therefore reduces the number of soldiers on the board. The Solitaire army is played on the infinite plane and the holes are in the lattice points Z × Z. The permitted moves are to jump horizontally or vertically. Suppose we start with all soldiers in the negative half plane. How many soldiers are needed to send one soldier forwards 1,2,3,4 or 5 holes into the upper plane? Obviously, 2 men are enough for sending one soldier 1 step forwards. Indeed, starting with the two soldiers standing on the same vertical line consecutively, the \lower" one can jump over the other and move 1 step forwards. It is also relatively easy to see that 4 soldiers are enough to send one 2 holes forwards (try it!). Less obvious is that eight soldiers are enough to push one soldier 3 holes forwards and 20 are enough to push a soldier 4 holes (exercise!). Before reading the next theorem, try to test your intuition by guessing the answer for the following question: Question 2.1. How many soldiers are needed to send one of them 5 holes forwards? To answer this question, probably the first thing I would do is to look at the online encyclopedia for integer sequences at https://oeis.org/. Based on that, the first search result which does not start at 1 gives us 52. It won't be very surprising if the answer is not 52 but some other, relatively small, number. But it is very surprising to know that: Theorem 2.2 (Conway, 1961). Impossible! Proof. The proof of this theorem pioneered the use of potential functions in combinatorics. The basic plan is quite simple: We wish to assign a weight to each hole with the condition that if ABC are three consecutive holes on a vertical/horizontal line with !(A);!(B);!(C) being their weights, then !(A) + !(B) ≥ !(C): (1) Suppose we have such a \weigh" function ! : Z × Z ! R, we evaluate a position in the game by the sum of all weights of holes which are occupied by soldiers { this sum is called the value of the position. Note that by 1 we have that each legal move cannot increase the value (prove it formally!). Therefore, such a weigh function proves that no matter how we paly, we can never get into a position of larger value from the initial position. The desired punch line will be to find such a function ! for which the position of having a soldier 5 steps forwards is larger than any initial position. 2 p Sounds easy? Let's do it! Let ! be a positive number satisfying ! + ! = 1 (the golden section { 5−1 2 ). Now, assume that we have managed to find a configuration of finitely many soldiers for which one soldier can move 5 holes forwards into the upper plane. Assign a weight 1 to the this hole in the positive upper plane, and extend it as follows: on the vertical line with the 1 on it downwards assign the weights 1; w; w2;:::. For each i ≥ 0, expand the horizontal line with !i to the left and right in a symmetric way by assigning the weights :::!i+`;!i+`−1;:::;!i+1; !i;!i+1;:::;!i+`−1;!i+`;:::. Now, a simple calculation gives us that the value of the top line of the lower half plane is w5 + 2w6 + 2w7 + ::: = w2: 2 P1 i 2 1 Therefore, the value of the whole lower half plane is i=2 ! = ! · 1−! = 1. In particular, since no hole in the lower plane has weight 0, no finite number of soldiers will suffice to send a man 5 forward! This completes the proof. 2.2 Strategy Stealing Strategy stealing is an existence argument which is oftenly used to determine the winner in a game between two perfect players. Unfortunately, it gives us no clue for how a winning/drawing strategy should look like. The strategy stealing argument is based on a symmetry argument, and therefore it works for symmetric games (whatever it means...). The class of games we are interested at in these notes are the so-called Positional Games. In a positional game there are two players, I and II, alternating turns in claiming previously unclaimed elements of some board V , with I going first. There is also a predetermined family of subsets of V , denoted by F which is considered as the collection of winning sets (the pair (V; F) is considered as the hypergraph of the game). The winner of the game is the first player to fully occupy all the elements of one of the winning sets in F. If there is no winner by the time that all the elements in V have been previously claimed, the game is declared as a draw. A complicated definition? not so... let us illustrate the definition by a simple example, namely the traditional child-game Tic-Tac-Toe. This game is played on a 3 × 3 board, where the players alternate turns in marking 0=X in previously unmarked spots. The winner is the first player to fully mark some combinatorial line (that is, horizontal, vertical, or diagonal line). This game can be easily described as a positional game! (do it). It sounds quite intuitive that the first player to move always have an advantage, and this is proven formally by the strategy stealing argument: Theorem 2.3 (Strategy stealing). Let (V; F) be any hypergraph of a positional game. Then, the first player can force at least a draw. Sketch of proof. Assume that the second player has a winning strategy and we wish to obtain a contradiction. The basic idea is to analyze what happens if the first player steals this strategy and plays accordingly. A winning strategy is a list of instructions telling the player how to answer each move. Now, Player I can play as follows: first move { arbitrary. From now on, she pretends to be the second player (and ignores her first move). If at some point she needs to claim the element she has claimed first, then she plays arbitrarily. The crucial observation here is that an extra move cannot harm any of the players. Let us now consider the following example: Consider a game with a board V = E(Kn) (that is, the board elements are all the edges of a complete graph on n vertices). Suppose that the winning sets consists of all subsets of edges that form a copy of Kk in Kn. We refer this game as the k-clique game played on E(Kn). A famous theorem of Ramsey asserts: Theorem 2.4 (Ramsey theorem). For every k there exists R(k) for which the following holds: every graph on at n ≥ R(k) many vertices either contains a copy of Kk or its complement (that is, the subgraph of Kn that consists of all the non-edges of G) contains a copy of Kk.